Geometry Vocabulary Cards
High School Mathematics
Geometry Vocabulary Word Wall Cards
Table of Contents
Reasoning, Lines, and
Transformations
Basics of Geometry 1
Basics of Geometry 2
Geometry Notation
Logic Notation
Set Notation
Conditional Statement
Converse
Inverse
Contrapositive
Symbolic Representations
Deductive Reasoning
Inductive Reasoning
Proof
Properties of Congruence
Law of Detachment
Law of Syllogism
Counterexample
Perpendicular Lines
Parallel Lines
Skew Lines
Transversal
Corresponding Angles
Alternate Interior Angles
Alternate Exterior Angles
Consecutive Interior Angles
Parallel Lines
Midpoint
Midpoint Formula
Slope Formula
Slope of Lines in Coordinate Plane
Distance Formula
Line Symmetry
Point Symmetry
Rotation (Origin)
Reflection
Translation
Dilation
Rotation (Point)
Perpendicular Bisector
Constructions:
o A line segment congruent to a given line segment
o Perpendicular bisector of a line segment
o A perpendicular to a given line from a point not on the line
o A perpendicular to a given line at a point on the line
o A bisector of an angle
o An angle congruent to a given angle
o A line parallel to a given line through a point not on the given line
o An equilateral triangle inscribed in a circle
o A square inscribed in a circle
o A regular hexagon inscribed in a circle
o An inscribed circle of a triangle
o A circumscribed circle of a triangle
o A tangent line from a point outside a given circle to the circle
Triangles
Classifying Triangles by Sides
Classifying Triangles by Angles
Triangle Sum Theorem
Exterior Angle Theorem
Pythagorean Theorem
Angle and Sides Relationships
Triangle Inequality Theorem
Congruent Triangles
SSS Triangle Congruence Postulate
SAS Triangle Congruence Postulate
HL Right Triangle Congruence
ASA Triangle Congruence Postulate
AAS Triangle Congruence Theorem
Similar Polygons
Similar Triangles and Proportions
AA Triangle Similarity Postulate
SAS Triangle Similarity Theorem
SSS Triangle Similarity Theorem
Altitude of a Triangle
Median of a Triangle
Concurrency of Medians of a Triangle
30°-60°-90° Triangle Theorem
45°-45°-90° Triangle Theorem
Geometric Mean
Trigonometric Ratios
Inverse Trigonometric Ratios
Area of a Triangle
Polygons and Circles
Polygon Exterior Angle Sum Theorem
Polygon Interior Angle Sum Theorem
Regular Polygon
Properties of Parallelograms
Rectangle
Rhombus
Square
Trapezoids
Circle
Circles
Circle Equation
Lines and Circles
Secant
Tangent
Central Angle
Measuring Arcs
Arc Length
Secants and Tangents
Inscribed Angle
Area of a Sector
Inscribed Angle Theorem 1
Inscribed Angle Theorem 2
Inscribed Angle Theorem 3
Segments in a Circle
Segments of Secants Theorem
Segment of Secants and Tangents Theorem
Three-Dimensional Figures
Cone
Cylinder
Polyhedron
Similar Solids Theorem
Sphere
Pyramid
Reasoning, Lines, and Transformations
Basics of Geometry
Point – A point has no dimension.
It is a location on a plane. It is
represented by a dot.
Line – A line has one dimension. It is an infinite set of points represented by a line with two arrowheads that extends without end.
Plane – A plane has two dimensions extending without end. It is often represented by a parallelogram.
Basics of Geometry
Line segment – A line segment consists of two endpoints and all the points between them.
Ray – A ray has one endpoint and extends without end in one direction.
Geometry Notation
Symbols used to represent statements or operations in geometry.
| |segment BC |
|BC | |
| |ray BC |
|BC | |
| |line BC |
|BC | |
|BC |length of BC |
|([pic] |angle ABC |
|m([pic] |measure of angle ABC |
|[pic] |triangle ABC |
||| |is parallel to |
|( |is perpendicular to |
|( |is congruent to |
|( |is similar to |
Logic Notation
|⋁ |or |
|⋀ |and |
|→ |read “implies”, if… then… |
|↔ |read “if and only if” |
|iff |read “if and only if” |
|~ |not |
|∴ |therefore |
Set Notation
|{} |empty set, null set |
|∅ |empty set, null set |
|x | |read “x such that” |
|x : |read “x such that” |
|⋃ |union, disjunction, or |
|⋂ |intersection, conjunction, and |
Conditional Statement
a logical argument consisting of
a set of premises,
hypothesis (p), and conclusion (q)
Symbolically:
if p, then q p(q
Converse
formed by interchanging the hypothesis and conclusion of a conditional statement
Conditional: If an angle is a right angle, then its measure is 90(.
Symbolically:
if q, then p q(p
Inverse
formed by negating the hypothesis and conclusion of a conditional statement
Conditional: If an angle is a right angle, then its measure is 90(.
Symbolically:
if ~p, then ~q ~p(~q
Contrapositive
formed by interchanging and negating the hypothesis and conclusion of a conditional statement
Conditional: If an angle is a right angle, then its measure is 90(.
Symbolically:
if ~q, then ~p ~q(~p
Symbolic Representations
|Conditional |if p, then q |p(q |
|Converse |if q, then p |q(p |
|Inverse |if not p, then not q |~p(~q |
|Contrapositive |if not q, then not p |~q(~p |
Deductive Reasoning
method using logic to draw conclusions based upon definitions, postulates, and theorems
Inductive Reasoning
method of drawing conclusions from a limited set of observations
Proof
a justification logically valid and based on initial assumptions, definitions, postulates, and theorems
Properties of Congruence
|Reflexive Property |For all angles A, (A ( (A. |
| |An angle is congruent to itself. |
|Symmetric Property |For any angles A and B, |
| |If (A ( (B, then (B ( (A . |
| |Order of congruence does not matter. |
|Transitive Property |For any angles A, B, and C, |
| |If (A ( (B and (B ( (C, then (A ( (C. |
| |If two angles are both congruent to a third angle, then the first two angles are also congruent. |
Law of Detachment
deductive reasoning stating that if the hypothesis of a true conditional statement is true then the conclusion is also true
Example:
If m(A > 90°, then (A is an obtuse angle. m(A = 120(.
Therefore, (A is an obtuse angle.
If p(q is a true conditional statement and p is true, then q is true.
Law of Syllogism
deductive reasoning that draws a new conclusion from two conditional statements when the conclusion of one is the hypothesis of the other
Example:
1. If a rectangle has four equal side lengths, then it is a square.
2. If a polygon is a square, then it is a regular polygon.
3. If a rectangle has four equal side lengths, then it is a regular polygon.
If p(q and q(r are true conditional statements, then p(r is true.
Counterexample
specific case for which a conjecture is false
One counterexample proves a conjecture false.
Perpendicular Lines
two lines that intersect to form a right angle
Line m is perpendicular to line n.
m ( n
Parallel Lines
lines that do not intersect and are coplanar
[pic]
m||n
Line m is parallel to line n.
Parallel lines have the same slope.
Skew Lines
lines that do not intersect and are not coplanar
Transversal
a line that intersects at least two other lines
Line t is a transversal.
Corresponding Angles
angles in matching positions when a transversal crosses at least two lines
Alternate Interior Angles
angles inside the lines and on opposite sides of the transversal
Alternate Exterior Angles
angles outside the two lines and on opposite sides of the transversal
Consecutive Interior Angles
angles between the two lines and on the same side of the transversal
Parallel Lines
Line a is parallel to line b when
|Corresponding angles are congruent |(1 ( (5, (2 ( (6, |
| |(3 ( (7, (4 ( (8 |
|Alternate interior angles are congruent |(3 ( (6 |
| |(4 ( (5 |
|Alternate exterior angles are congruent |(1 ( (8 |
| |(2 ( (7 |
|Consecutive interior angles are supplementary |m(3+ m(5 = 180° |
| |m(4 + m(6 = 180° |
Midpoint
divides a segment into two congruent segments
Example: M is the midpoint of CD
CM ( MD
CM = MD
Segment bisector may be a point, ray, line, line segment, or plane that intersects the segment at its midpoint.
Midpoint Formula
given points A(x1, y1) and B(x2, y2)
midpoint M =
Slope Formula
ratio of vertical change to
horizontal change
|slope |= |m |= |change in y |= |y2 – y1 |
| | | | |change in x | |x2 – x1 |
Slopes of Lines
Distance Formula
given points A (x1, y1) and B (x2, y2)
Line Symmetry
MOM
B X
Point Symmetry
pod
S Z
Rotation
|Preimage |Image |
|A(-3,0) |A((0,3) |
|B(-3,3) |B((3,3) |
|C(-1,3) |C((3,1) |
|D(-1,0) |D((0,1) |
Pre-image has been transformed by a 90( clockwise rotation about the origin.
Rotation
Pre-image A has been transformed by a 90( clockwise rotation about the point (2, 0) to form image AI.
Reflection
[pic]
|Preimage |Image |
|D(1,-2) |D((-1,-2) |
|E(3,-2) |E((-3,-2) |
|F(3,2) |F((-3,2) |
Translation
[pic]
|Preimage |Image |
|A(1,2) |A((-2,-3) |
|B(3,2) |B((0,-3) |
|C(4,3) |C((1,-2) |
|D(3,4) |D((0,-1) |
|E(1,4) |E((-2,-1) |
Dilation
|Preimage |Image |
|A(0,2) |A((0,4) |
|B(2,0) |B((4,0) |
|C(0,0) |C((0,0) |
|Preimage |Image |
|E |E( |
|F |F( |
|G |G( |
|H |H( |
Perpendicular
Bisector
a segment, ray, line, or plane that is perpendicular to a segment at its midpoint
Example:
Line s is perpendicular to XY.
M is the midpoint, therefore XM ( MY.
Z lies on line s and is equidistant from X and Y.
Constructions
Traditional constructions involving a compass and straightedge reinforce students’ understanding of geometric concepts. Constructions help students visualize Geometry.
There are multiple methods to most geometric constructions. These cards illustrate only one method. Students would benefit from experiences with more than one method and should be able to justify each step of geometric constructions.
Construct
segment CD congruent to segment AB
Construct
a perpendicular bisector of segment AB
Construct
a perpendicular to a line from point P not on the line
Construct
a perpendicular to a line from point P on the line
Construct
a bisector of (A
Construct
(Y congruent to (A
Construct
line n parallel to line m through point P not on the line
Construct
an equilateral triangle inscribed
in a circle
Construct
a square inscribed in a circle
Construct
a regular hexagon inscribed
in a circle
Construct
the inscribed circle of a triangle
Construct
the circumscribed circle
of a triangle
Construct
a tangent from a point outside a given circle to the circle
Triangles
Classifying Triangles
|Scalene |Isosceles |Equilateral |
| | | |
|No congruent sides |At least 2 congruent sides |3 congruent sides |
|No congruent angles |2 or 3 congruent angles |3 congruent angles |
All equilateral triangles are isosceles.
Classifying Triangles
|Acute |Right |Obtuse |Equiangular |
| | | | |
|3 acute angles |1 right angle |1 obtuse angle |3 congruent angles |
|3 angles, each less than 90( |1 angle equals 90( |1 angle greater than 90( |3 angles, |
| | | |each measures 60( |
Triangle Sum Theorem
measures of the interior angles of a triangle = 180(
m(A + m(B + m(C = 180(
Exterior Angle Theorem
Exterior angle, m(1, is equal to the sum of the measures of the two nonadjacent interior angles.
m(1 = m(B + m(C
Pythagorean Theorem
If (ABC is a right triangle, then
a2 + b2 = c2.
Conversely, if a2 + b2 = c2, then
(ABC is a right triangle.
Angle and Side Relationships
(A is the largest angle,
therefore BC is the
longest side.
(B is the smallest angle, therefore AC is the shortest side.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Example:
AB + BC > AC AC + BC > AB
AB + AC > BC
Congruent Triangles
Two possible congruence statements:
(ABC ( (FED
(BCA ( (EDF
Corresponding Parts of Congruent Figures
|(A ( (F |AB ( FE |
|(B ( (E |BC ( ED |
|(C ( (D |CA ( DF |
SSS Triangle Congruence Postulate
Example:
If Side AB ( FE,
Side AC ( FD, and
Side BC ( ED ,
then ( ABC ( (FED.
SAS Triangle Congruence Postulate
Example:
If Side AB ( DE,
Angle (A ( (D, and
Side AC ( DF ,
then ( ABC ( (DEF.
HL Right Triangle Congruence
Example:
If Hypotenuse RS ( XY, and
Leg ST ( YZ ,
then ( RST ( (XYZ.
ASA Triangle Congruence Postulate
Example:
If Angle (A ( (D,
Side AC ( DF , and
Angle (C ( (F
then ( ABC ( (DEF.
AAS Triangle Congruence Theorem
Example:
If Angle (R ( (X,
Angle (S ( (Y, and
Side ST ( YZ
then ( RST ( (XYZ.
Similar Polygons
|ABCD ( HGFE |
|Angles |Sides |
|(A corresponds to (H |[pic] corresponds to [pic] |
|(B corresponds to (G |[pic] corresponds to [pic] |
|(C corresponds to (F |[pic] corresponds to [pic] |
|(D corresponds to (E |[pic] corresponds to [pic] |
Corresponding angles are congruent.
Corresponding sides are proportional.
Similar Polygons and Proportions
Corresponding vertices are listed in the same order.
Example: (ABC ( (HGF
[pic] = [pic]
[pic] = [pic]
The perimeters of the polygons are also proportional.
AA Triangle Similarity Postulate
Example:
If Angle (R ( (X and
Angle (S ( (Y,
then (RST ( (XYZ.
SAS Triangle Similarity Theorem
Example:
If (A ( (D and
[pic] = [pic]
then (ABC ( (DEF.
SSS Triangle Similarity Theorem
Example:
If [pic] = [pic] = [pic]
then (RST ( (XYZ.
Altitude of a Triangle
a segment from a vertex perpendicular to the opposite side
Every triangle has 3 altitudes.
The 3 altitudes intersect at a point called the orthocenter.
Median of a Triangle
D is the midpoint of AB; therefore, CD is a median of (ABC.
Every triangle has 3 medians.
Concurrency of Medians of a Triangle
Medians of (ABC intersect at P and
AP = [pic]AF, CP = [pic]CE , BP = [pic]BD.
30°-60°-90° Triangle
Theorem
Given: short leg = x
Using equilateral triangle,
hypotenuse = 2 ∙ x
Applying the Pythagorean Theorem,
longer leg = x ∙[pic]
45°-45°-90° Triangle
Theorem
Given: leg = x,
then applying the Pythagorean Theorem;
hypotenuse2 = x2 + x2
hypotenuse = x[pic]
Geometric Mean
of two positive numbers a and b is the positive number x that satisfies
[pic] = [pic].
x2 = ab and x = [pic].
In a right triangle, the length of the altitude is the geometric mean of the lengths of the two segments.
Example:
= , so x2 = 36 and x = [pic] = 6.
Trigonometric
Ratios
sin A = =
cos A = =
tan A = =
Inverse Trigonometric Ratios
|Definition |Example |
|If tan A = x, then tan-1 x = m(A. |tan-1 [pic] = m(A |
|If sin A = y, then sin-1 y = m(A. |sin-1 [pic]= m(A |
|If cos A = z, then cos-1 z = m(A. |cos-1 [pic] = m(A |
Area of Triangle
sin C = [pic]
h = a∙sin C
A = [pic]bh (area of a triangle formula)
By substitution, A = [pic]b(a∙sin C)
A = [pic]ab∙sin C
Polygons and Circles
Polygon Exterior Angle Sum Theorem
The sum of the measures of the exterior angles of a convex polygon is 360°.
Example:
m(1 + m(2 + m(3 + m(4 + m(5 = 360(
Polygon Interior Angle Sum Theorem
The sum of the measures of the interior angles of a convex n-gon is (n – 2)∙180°.
S = m(1 + m(2 + … + m(n = (n – 2)∙180°
Example:
If n = 5, then S = (5 – 2)∙180°
S = 3 ∙ 180° = 540°
Regular Polygon
a convex polygon that is both equiangular and equilateral
Properties of Parallelograms
• Opposite sides are parallel and congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• The diagonals bisect each other.
Rectangle
• A rectangle is a parallelogram with four right angles.
• Diagonals are congruent.
• Diagonals bisect each other.
Rhombus
• A rhombus is a parallelogram with four congruent sides.
• Diagonals are perpendicular.
• Each diagonal bisects a pair of opposite angles.
Square
• A square is a parallelogram and a rectangle with four congruent sides.
• Diagonals are perpendicular.
• Every square is a rhombus.
Trapezoid
• A trapezoid is a quadrilateral with exactly one pair of parallel sides.
• Isosceles trapezoid – A trapezoid where the two base angles are equal and therefore the sides opposite the base angles are also equal.
Circle
all points in a plane equidistant from a given point called the center
• Point O is the center.
• MN passes through the center O and therefore, MN is a diameter.
• OP, OM, and ON are radii and
OP ( OM ( ON.
• RS and MN are chords. [pic]
Circles
A polygon is an inscribed polygon if all of its vertices lie on a circle.
A circle is considered
“inscribed” if it is
tangent to each side
of the polygon.
Circle Equation
x2 + y2 = r2
circle with radius r and center at
the origin
standard equation of a circle
(x – h)2 + (y – k)2 = r2
with center (h,k) and radius r
Lines and Circles
• Secant (AB) – a line that intersects a circle in two points.
• Tangent (CD) – a line (or ray or segment) that intersects a circle in exactly one point, the point of tangency, D.
Secant
If two lines intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measures of the intercepted arcs.
m(1 = [pic](x° + y°)
Tangent
A line is tangent to a circle if and only if the line is perpendicular to a radius drawn to the point of tangency.
QS is tangent to circle R at point Q.
Radius RQ ( QS
Tangent
If two segments from the same exterior point are tangent to a circle, then they are congruent.
AB and AC are tangent to the circle
at points B and C.
Therefore, AB ( AC and AC = AB.
Central Angle
an angle whose vertex is the center of the circle
[pic]
(ACB is a central angle of circle C.
Minor arc – corresponding central angle is less than 180°
Major arc – corresponding central angle is greater than 180°
Measuring Arcs
|Minor arcs |Major arcs |Semicircles |
|[pic] |[pic] |[pic] |
|[pic] |[pic] |[pic] |
The measure of the entire circle is 360o.
The measure of a minor arc is equal to its central angle.
The measure of a major arc is the difference between 360° and the measure of the related minor arc.
Arc Length
Example:
Secants and Tangents
m(1 = [pic](x°- y°)
Inscribed Angle
angle whose vertex is a point on the circle and whose sides contain chords of the circle
[pic]
Area of a Sector
region bounded by two radii and their intercepted arc
Inscribed Angle Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
(BDC ( (BAC
Inscribed Angle Theorem
m(BAC = 90° if and only if BC is a diameter of the circle.
Inscribed Angle Theorem
M, A, T, and H lie on circle J if and only if
m(A + m(H = 180° and
m(T + m(M = 180°.
Segments in a Circle
If two chords intersect in a circle,
then a∙b = c∙d.
Example:
12(6) = 9x
72 = 9x
8 = x
Segments of Secants Theorem
AB ∙ AC = AD ∙ AE
Example:
6(6 + x) = 9(9 + 16)
36 + 6x = 225
x = 31.5
Segments of Secants and Tangents Theorem
AE2 = AB ∙ AC
Example:
252 = 20(20 + x)
625 = 400 + 20x
x = 11.25
Three-Dimensional Figures
Cone
solid that has a circular base, an apex, and a lateral surface
Cylinder
solid figure with congruent circular bases that lie in parallel planes
Polyhedron
solid that is bounded by polygons, called faces
[pic] [pic]
Similar Solids Theorem
If two similar solids have a scale factor of a:b, then their corresponding surface areas have a ratio of a2: b2, and their corresponding
volumes have a ratio of a3: b3.
cylinder A ( cylinder B
|Example |
|scale factor |a : b |3:2 |
|ratio of |a2: b2 |9:4 |
|surface areas | | |
|ratio of volumes |a3: b3 |27:8 |
Sphere
a three-dimensional surface of which all points are equidistant from
a fixed point
Pyramid
polyhedron with a polygonal base and triangular faces meeting in a common vertex
-----------------------
P
point P
AB or BA or line m
A B
m
N
A
B
C
plane ABC or plane N
AB or BA
B
A
C
BC
Note: Name the endpoint first.
BC and CB are different rays.
B
hypothesis
If an angle is a right angle,
then its measure is 90(.
conclusion
Converse: If an angle measures 90(, then the angle is a right angle.
Inverse: If an angle is not a right angle, then its measure is not 90(.
Converse: If an angle does not measure 90(, then the e does not measure 90(, then the angle is not a right angle.
Example:
Prove (x ∙ y) ∙ z = (z ∙ y) ∙ x.
Step 1: (x ∙ y) ∙ z = z ∙ (x ∙ y), using commutative property of multiplication.
Step 2: = z ∙ (y ∙ x), using commutative property of multiplication.
Step 3: = (z ∙ y) ∙ x, using associative property of multiplication.
Example:
Given a pattern, determine the rule for the pattern.
Determine the next number in this sequence 1, 1, 2, 3, 5, 8, 13...
Example:
Given: (1 ( (2
Prove: (2 ( (1
|Statements |Reasons |
|(1 ( (2 |Given |
|m(1 = m(2 |Definition of congruent angles |
|m(2 = m(1 |Symmetric Property of Equality |
|(2 ( (1 |Definition of congruent angles |
A
120(
Example:
Conjecture: “The product of any two numbers is odd.”
Counterexample: 2 ∙ 3 = 6
m
n
m
n
m
n
t
x
y
t
b
a
t
a
b
4
5
6
3
2
1
7
8
Examples:
1) (2 and (6
2) (3 and (7
a
b
t
2
3
4
1
Examples:
1) (1 and (4
2) (2 and (3
t
a
b
2
1
3
4
Examples:
1) (1 and (4
2) (2 and (3
2
1
3
4
t
a
b
Examples:
1) (1 and (2
2) (3 and (4
a
b
t
4
5
6
3
2
1
7
8
D
C
M
A
B
M
(x1, y1)
(x2, y2)
A
B
(x1, y1)
(x2, y2)
x2 – x1
y2 – y1
Parallel lines have the same slope.
Perpendicular lines have slopes whose product is -1.
Vertical lines have undefined slope.
Horizontal lines have 0 slope.
y
x
n
p
Example:
The slope of line n = -2. The slope of line p =[pic].
-2 ∙ [pic] = -1, therefore, n ( p.
[pic]
A
B
(x1, y1)
(x2, y2)
x2 – x1
y2 – y1
The distance formula is based on the Pythagorean Theorem.
A
Aˊ
C
Cˊ
P
x
y
A(
D
B
C
A
B(
C(
D(
center of rotation
[pic]
center of rotation
x
A
A'
y
y
x
D
F
E
D(
E(
F(
y
E(
D
A
B
E
C
C(
D(
A(
B(
x
C
A
B
A(
B(
C(
y
x
E
F
G
P
E(
F(
H(
H
G(
s
Z
Y
X
M
B
A
C
D
Fig. 2
Fig. 1
B
A
B
A
Fig. 2
Fig. 1
Fig. 3
A
B
P
Fig. 3
B
A
B
A
P
Fig. 4
Fig. 1
Fig. 2
B
A
P
B
A
P
P
B
A
P
B
A
P
Fig. 2
B
A
Fig. 4
Fig. 3
Fig. 1
B
A
P
A
A
Fig. 4
Fig. 3
Fig. 1
Fig. 2
A
A
A
Y
A
A
A
Y
Y
Y
Fig. 4
Fig. 3
Fig. 1
Fig. 2
m
P
m
P
Fig. 2
Fig. 1
Draw a line through point P intersecting line m.
Fig. 4
Fig. 3
m
P
m
P
n
Fig. 4
Fig. 3
Fig. 1
Fig. 2
Fig. 2
Fig. 1
Draw a diameter.
Fig. 3
Fig. 4
Fig. 4
Fig. 3
Fig. 1
Fig. 2
Bisect all angles.
Fig. 3
Fig. 1
Fig. 2
Fig. 4
Fig. 1
Fig. 3
Fig. 2
Fig. 4
P
P
P
P
Fig. 4
Fig. 3
Fig. 1
Fig. 2
B
A
C
A
B
C
1
b
c
hypotenuse
a
B
A
C
A
12
8
6
88o
54o
38o
B
C
12
8
6
88o
54o
38o
B
C
A
A
B
C
A
B
C
F
D
E
A
B
C
F
D
E
A
B
C
F
E
D
R
S
T
X
Y
Z
B
C
F
E
D
A
R
S
T
X
Y
Z
A
B
D
C
E
F
G
H
2
4
6
12
A
B
C
H
G
F
12
6
4
x
R
S
T
X
Y
Z
12
6
14
7
F
E
D
A
B
C
Y
S
2.5
6.5
5
13
Z
X
T
R
6
12
G
J
H
altitudes
orthocenter
altitude/height
B
C
A
a
D
median
A
C
B
A
B
C
D
E
F
centroid
P
30°
60°
x
2x
x[pic]
x
60°
30°
x
x
x[pic][pic][pic]
45°
45°
A
C
B
x
9
4
(side adjacent (A)
A
B
C
a
b
c
(side opposite (A)
(hypotenuse)
a
c
hypotenuse
side opposite (A
b
c
hypotenuse
side adjacent (A
a
b
side adjacent to (A
side opposite (AA
A
B
C
a
b
c
h
A
B
C
a
b
5
2
3
4
1
5
2
3
4
1
Equilateral Triangle
Each angle measures 60o.
Square
Each angle measures 90o.
Regular Pentagon
Each angle measures 108o.
Regular Hexagon
Each angle measures 120o.
Regular Octagon
Each angle measures 135o.
y
x
(x,y)
x
y
r
C
D
A
B
y°
1
x°
Q
S
R
C
B
A
A
B
C
minor arc AB
major arc ADB
D
D
B
R
C
70°
110°
A
4 cm
A
B
C
120°
Two secants
1
x°
y°
Secant-tangent
1
x°
y°
Two tangents
1
x°
y°
B
A
C
cm
Example:
A
D
B
C
O
A
C
B
88(
92(
95(
85(
M
J
T
H
A
92(
85(
a
b
c
d
12
6
x
9
B
A
C
D
E
9
6
x
16
A
B
C
E
25
20
x
apex
slant height (l)
lateral surface
(curved surface of cone)
radius(r)
height (h)
base
V = [pic](r2h
L.A. (lateral surface area) = (rl
S.A. (surface area) = (r2 + (rl
height (h)
radius (r)
base
base
V = (r2h
L.A. (lateral surface area) = 2(rh
S.A. (surface area) = 2(r2 + 2(rh
A
B
radius
V = [pic](r3
S.A. (surface area) = 4(r2
vertex
base
slant height (l)
height (h)
area of base (B)
perimeter of base (p)
V = [pic]Bh
L.A. (lateral surface area) = [pic]lp
S.A. (surface area) = [pic]lp + B
................
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